Problem: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $n \neq 0$. $y = \dfrac{-3}{5(5n + 4)} \div \dfrac{2n}{n(5n + 4)} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $y = \dfrac{-3}{5(5n + 4)} \times \dfrac{n(5n + 4)}{2n} $ When multiplying fractions, we multiply the numerators and the denominators. $y = \dfrac{ -3 \times n(5n + 4) } { 5(5n + 4) \times 2n } $ $ y = \dfrac{-3n(5n + 4)}{10n(5n + 4)} $ We can cancel the $5n + 4$ so long as $5n + 4 \neq 0$ Therefore $n \neq -\dfrac{4}{5}$ $y = \dfrac{-3n \cancel{(5n + 4})}{10n \cancel{(5n + 4)}} = -\dfrac{3n}{10n} = -\dfrac{3}{10} $